Single-Qubit Quantum Gates on a Bloch Sphere
Using the Bloch sphere, a cubit can be represented as a unit vector (shown in red) from the origin to the point on the unit sphere with spherical coordinates . A single-qubit quantum gate operating on produces a rotated qubit , represented by the green vector. Check the box for "add gate 2?" to perform a second operation using gate . This produces another qubit , which is represented by the blue vector. You can choose from the gates H, X, Y, Z, S and T as defined in the Details.
The action of the single-qubit quantum gates can be represented by unitary matrices acting on the qubit
Hadamard gate: TextData[""]=(111-1),
Pauli X gate: InlineMath=(0110),
Pauli Y gate: TextData[""]=(0-ii0),
Pauli Z gate: InlineMath=(100-1),
phase (or gate: InlineMath=(100i),
gate: InlineMath=(100ei π4).
In general, the operation InlineMath (αβ) gives a qubit of the form
with an overall phase factor of no physical significance. Multiplication by removes this factor and reduces the qubit to the canonical form
 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge: Cambridge University Press, 2010.
 G. Fano and S. M. Blinder, Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers, Berlin: Springer, 2017.